I think recursively enumerable is sufficient but not necessary. To see an example, consider Peano Arithmetic with added axioms as follows: Pick some lexographic ordering of all well-formed formula in PA, and denote this system by P_0. Define Pn for n>1 by running through your list of statements until you come to one that isn’t provable or disprovable in P(n-1). When you do, throw it in as true with the axioms of P_(n-1) to get P_n. Consider then the system P_infinity = the union of all the P_i. This system has only finitely many axioms, is Godel numerable by most definitions of that term but is not recursively enumerable (since if it were we’d have a decision procedure for PA.)
Yes, you’re right, of course. In my above comment, I failed to consider that not only Turing machines can be Goedelized, but also various infinite-step procedures that produce non-computable results, such as the one you outlined above.
(Also, I assume you meant “countably,” not “finitely” in the last sentence.)
I think recursively enumerable is sufficient but not necessary. To see an example, consider Peano Arithmetic with added axioms as follows: Pick some lexographic ordering of all well-formed formula in PA, and denote this system by P_0. Define Pn for n>1 by running through your list of statements until you come to one that isn’t provable or disprovable in P(n-1). When you do, throw it in as true with the axioms of P_(n-1) to get P_n. Consider then the system P_infinity = the union of all the P_i. This system has only finitely many axioms, is Godel numerable by most definitions of that term but is not recursively enumerable (since if it were we’d have a decision procedure for PA.)
Yes, you’re right, of course. In my above comment, I failed to consider that not only Turing machines can be Goedelized, but also various infinite-step procedures that produce non-computable results, such as the one you outlined above.
(Also, I assume you meant “countably,” not “finitely” in the last sentence.)